The Gromov-Wasserstein (GW) distance quantifies discrepancy between metric measure spaces, but suffers from computational hardness. The entropic Gromov-Wasserstein (EGW) distance serves as a computationally efficient proxy for the GW distance. Recently, it was shown that the quadratic GW and EGW distances admit variational forms that tie them to the well-understood optimal transport (OT) and entropic OT (EOT) problems. By leveraging this connection, we derive two notions of stability for the EGW problem with the quadratic or inner product cost. The first stability notion enables us to establish convexity and smoothness of the objective in this variational problem. This results in the first efficient algorithms for solving the EGW problem that are subject to formal guarantees in both the convex and non-convex regimes. The second stability notion is used to derive a comprehensive limit distribution theory for the empirical EGW distance and, under additional conditions, asymptotic normality, bootstrap consistency, and semiparametric efficiency thereof.

翻译：Gromov-Wasserstein (GW) 距离量化了度量测度空间之间的差异，但其计算复杂度较高。熵化Gromov-Wasserstein (EGW) 距离作为GW距离的一种高效计算替代方案，近年来研究表明，具有二次型或内积代价的GW和EGW距离均存在变分形式，将其与经典的最优传输(OT)及熵化最优传输(EOT)问题相关联。基于这一联系，我们推导了EGW问题在二次型或内积代价下的两种稳定性概念。第一种稳定性概念使我们能够建立该变分问题中目标函数的凸性与光滑性，从而首次提出在凸与非凸场景下均具有严格理论保证的EGW问题高效算法。第二种稳定性概念则用于推导经验EGW距离的完整极限分布理论，并在附加条件下进一步建立了其渐近正态性、自举一致性与半参数效率性质。